I have designed a small antenna. Now., I want to know how to determine the minimum gain for it according to Chu–Harrington limit Because this limit relates the size of the antenna to the Q-factor. But I want a way to find out the minimum gain of an antenna.
I think the Chu Harrington limit only describes the minimum Q of the antenna, not the achievable gain. The Q is determined by how much energy the antenna has to store before the currents are large enough to radiate all the input power. The gain is determined by how efficiently you can store all this energy, without losing it in conductor resistance, for example.
The maximum gain is determined by the losses in the tuning circuit or the antenna elements. If these are low then a gain close to 1 is possible.
There is a simple treatment by Wheeler that gives a useful result for the integral under the curve of efficiency plotted against the cube of wavelength, which is proportional to the volume of the enclosing sphere. This was derived with parameters from practical tuning circuits, using results from Bode and Fano. I haven't been able to locate the reference.
If you only want a narrow bandwidth, then a gain close to 1 is possible, if you can get a tuning circuit with a similar Q to the antenna or better, which is the limiting factor. It is hard to get a tuning circuit, or even an antenna, where the equivalent resistance of the antenna elements or tuning elements, transformed to be in series with the radiation resistance, isn't larger than the radiation resistance. It is this ratio that mainly determines the efficiency. The directivity is close to 1, as the antenna will radiate as a dipole or monopole.
You say minimum gain twice, but I expect you mean maximum gain as in the headline. The minimum gain is of course zero!
https://arxiv.org/ftp/arxiv/papers/1907/1907.11683.pdf
This isn't the method I referred to but may help!
Wheeler used the approximation mod(gamma) = 1-efficiency/2 which is close for small efficiencies, and then ln(1/(1-efficiency/2))=efficiency/2, so he plots efficiency rather than ln(1/mod(gamma)). If you can achieve good efficiency then the approximations are not valid and the paper above may be better for you!
According to Harrington (https://nvlpubs.nist.gov/nistpubs/jres/64D/jresv64Dn1p1_A1b.pdf)
G=N^2+2N
with N is the number of modes.
From this, the gain has potentially no limit if the number of modes can be increased. However, this fact is limited by the possibility to manufacture an antenna radiating many propagating modes, and losses because generally higher-order modes have higher losses.
In practice, the modes of order n>ka are rapidly cut off and are not naturally present in the field of an antenna of radius (a), and we consider N=ka
So, the maximum gain can be calculated by
G=(ka)^2+2ka
For more references, you can search for the work of Mats Gustafsson
If you read Harrington's paper you will find that the gain that is unlimited is called directive gain. Antenna engineers usually call this directivity, which describes the effect of beam width and shape but does not include the effect of losses. He refers to supergain, which antenna engineers usually call superdirectivity, which again does not include the effect of losses.
The correct definition of antenna gain is that it includes the effect of losses.
Near the end he includes and discusses the effect of losses and introduces a gain that does include the effect of losses.
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